Sepp Blatter has refused to resign as Fifa president following a request from Uefa chief Michel Platini.
The 79-year-old Swiss held an emergency meeting with key Fifa officials on Thursday after world football's governing body was subjected to yet more damaging corruption claims.
Platini then made a personal appeal for Blatter to quit but was told it was "too late" to resign.
Blatter later vowed to restore trust in Fifa and "find a way to fix things".
Opening a meeting of Fifa's congress on the eve of Friday's presidential election, he said: "These are unprecedented and difficult times for Fifa.
"It must fall to me to uphold responsibility for the well-being of the organisation."
However, while admitting he could not be held responsible for the actions of a few, he conceded: "I'm sure more bad news may follow but it's necessary to restore trust in our organisation."
His speech came just hours after Frenchman Platini said the latest crisis had left him "absolutely sickened", adding: "People have had enough, they don't want this president any more."
Blatter, who has been president since 1998, is seeking a fifth term when he takes on Prince Ali bin al-Hussein in Friday's election.
Several influential football figures had called for the vote to be delayed after seven Fifa officials were arrested in Zurich on Wednesday.
But Uefa, which governs European football, decided on Thursday not to boycott the election and will continue to back Prince Ali, although some member associations, such as Russia, have said they will back Blatter.
While Prince Ali has not given up on prising some votes from Africa, Caf, which looks after the interests of the continent, has reiterated its support for Blatter.
President of the Nigerian Football Federation, Amaju Pinnick, told BBC sports editor Dan Roan that Blatter will definitely win the election and he should not be held accountable for the corruption allegations.
The Asian Football Confederation, too, supports the Swiss - although one of its members, the Australian football federation, has announced its intention to vote for Prince Ali.
Fifa was plunged into fresh crisis on Wednesday when United States authorities indicted 14 people and arrested seven senior football officials on bribery and racketeering charges.
In a separate development, Swiss officials opening criminal proceedings into the bidding process for the 2018 and 2022 World Cup bid processes.
Blatter's decision to chair an emergency meeting with representatives from Fifa's six confederations is significant.
Such gatherings are rare.
It also took place without two of its nominated members.
Jeffrey Webb, president of the North, Central American and Caribbean Association (Concacaf), and Eugenio Figueredo, president of the South American confederation, were both absent after being arrested on Wednesday.
Concacaf later said it had provisionally suspended Webb and president of the Costa Rican Football Federation Eduardo Li, who was also among the 14 indicted by US authorities.
Blatter, who has been in power since 1998, was widely expected to win a fifth term as president before the current crisis engulfed Fifa.
But things are not so clear cut now.
He is understood to have widespread support among Fifa's 209 member associations, but Prince Ali, a Fifa vice-president from Jordan, could benefit from the latest crisis to his world football's governing body.
Prime Minister David Cameron and English Football Association chairman Greg Dyke joined those calling for Blatter to step down.
Britain's representative David Gill also says he will resign from Fifa's executive committee if Blatter is re-elected.
At their meeting on Thursday, Uefa delegates even discussed leaving Fifa and boycotting tournaments should Blatter be re-elected.
The Scottish Football Association's Stewart Regan told BBC Radio Scotland: "All of these things have been discussed."
But support for the president has come from Russian president Vladimir Putin, who has claimed the United States is meddling in Fifa's affairs in an attempt to take the 2018 World Cup away from his country.
It's not just about the election, is it?
No, it's not. Fifa's entire future appears to be at stake.
Its reputation has been sullied by corruption claims for years now.
However, there is a sense that these latest developments could have a seismic impact on the footballing landscape.
Let's remember, there are two investigations here.
One is led by the United States, which is focusing on bribery claims going back more than 20 years and involving several key Fifa figures.
The other, led by Swiss authorities, is focusing on potential wrongdoing when voting took place for 2018 and 2022 World Cup hosting rights.
How serious are the bribery claims?
Very. Seven Fifa officials were arrested in Zurich on Wednesday on charges they received more than $150m (£100m) in bribes.
Among them was Fifa vice-president Webb, a hugely influential figure who holds a lot of power in North America and the Caribbean.
In total, 14 defendants were charged by the US Department of Justice with racketeering, wire fraud and money laundering conspiracies in a 24-year scheme.
Jack Warner, a former Fifa vice-president who quit world football's governing body in 2011, was also among them.
The US is trying to extradite all seven officials arrested in Zurich. Unsurprisingly, all seven are fighting such a move.
What about the 2018 and 2022 World Cups?
It seems unlikely that either Russia or Qatar will be stripped of the tournaments at this stage, although anything is possible.
Despite Fifa's numerous attempts to prove the bidding process was fair, rumours persist that not everything was above board.
The latest development have only added to the speculation.
Naturally, both Russia, who were awarded the rights to stage the 2018 tournament, and Qatar, who will host the 2022 event, are fighting hard.
Both have always insisted that they won their bids fair and square.
Is the 2010 World Cup also under scrutiny?
Yes. The tournament was always going to be in Africa, but South Africa was chosen ahead of Egypt and Morocco.
However, the US investigation claims South African officials paid $10m (£6.5m) in bribes to host the tournament.
That has prompted a furious reaction from the South African government.
"When we concluded the Fifa World Cup here in South Africa, we got a clean audit report," said Minister in the Presidency Jeff Radebe. "There has never been any suggestion that anything untoward happened."
How is Fifa dealing with the crisis?
As you'd expect, it is working overtime to calm fears it is in meltdown.
Fifa director of communications Walter De Gregorio tried hard to put a positive spin on developments on Wednesday.
But not everyone believed him when he claimed the investigations proved Fifa was on the right track.
In an attempt to be proactive, Fifa has already banned 11 of the 14 people charged by the US Department of Justice.
But it is coming under increasing pressure from unhappy sponsors.
Which sponsors are particularly upset?
Visa. It says it will "reassess" its sponsorship unless Fifa takes "swift and immediate steps" to address the latest accusations.
Coca-Cola, Adidas, Nike and McDonalds have also voiced concern.
John Whittingdale, Britain's Secretary of State for Culture, Media and Sport, has urged all sponsors to "consider following Visa's lead".
It's fair to say that the loss of a high-profile sponsor would have a damaging impact on Fifa, both in terms of revenue and reputation.
As for World Cup broadcasters, most have not commented. Germany's ZDF/ARD told BBC Sport it had already signed binding contracts for the 2018 and 2022 tournaments, but added its journalists would continue to "report critically and seriously about the institutions of the sport".